How did a weighing scale end the fire-substance theory?

A weighing scale ended it. For about a century chemists explained burning with phlogiston, a fire-like element supposedly contained in anything combustible and released when it burned. Then Antoine Lavoisier weighed things before and after burning, in sealed vessels where no gas could escape, and found that they got heavier, because they combined with air. A substance escaping could not explain a gain in weight.

The theory was not foolish. It tried to explain combustion and rusting with one idea: a metal was a calx (a metal oxide) plus phlogiston, and igniting the metal freed the phlogiston and left the calx behind. But Stahl, who put the theory together in 1697, did not account for the increase in mass when tin and lead burn. Some followers even concluded that phlogiston had levity, making a compound lighter than it would be without it.

Lavoisier's answer was to count. In 1772 he reported that burning phosphorus combined with a large quantity of air and gained weight. In 1774 he burned tin and lead in sealed vessels and confirmed that their weight gain came from combining with air. He then named oxygen in 1778 and recognised it as an element.

His method mattered as much as his result. He weighed reactants and products together in a sealed glass vessel, and found the total mass the same at the end as at the beginning: burn wood to ashes, and the mass is unchanged once the gases are included. That is the law of conservation of mass, and precise measurements like his led to the balanced reaction equations still used today.